Dynamical systems and ergodic theory·OpenAI Math·Result 148·Lean formalisation listed
The entropy-rate dimension formula for self-similar measures
One of the 372 results in OpenAI's public mathematics catalogue, filed under dynamical systems and ergodic theory: 1 preprint, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a proof of the entropy-rate dimension formula for self-similar measures (dynamical systems and ergodic theory) in a preprint dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 11 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 3:28 · watch from there on YouTube →
What OpenAI claims
For every self-similar measure on the line generated by finitely many contracting similarities, proves dimHμ=min {1,hRW/χ}, where hRW is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios. (Lean)
Paper
The entropy-rate dimension formula for self-similar measures on the line
OpenAI's abstract We prove that the Hausdorff dimension of every finite real self-similar measure equals the minimum of one and its random-walk entropy rate divided by its Lyapunov exponent. Exact overlaps are allowed, and the contraction ratios may be unequal and negative. This resolves the entropy-rate dimension conjecture.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- For every self-similar measure on the line generated by finitely many contracting similarities, proves dimHμ=min {1,hRW/χ}, where hRW is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios. (Lean)
- Where is the proof?
- In the paper, The entropy-rate dimension formula for self-similar measures on the line (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 11).
Also in
- Chapter 4 of 9 in Birkhoff Conjecture, Zero Entropy and 7 More in Dynamical Systems and Ergodic Theory · starts at 1:04 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 11 from 3:28 · description, sources and chapters →
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